Extending group actions on metric spaces

We address the following natural extension problem for group actions: Given a group $G$, a subgroup $H\le G$, and an action of $H$ on a metric space, when is it possible to extend it to an action of the whole group $G$ on a (possibly different) metric space? When does such an extension preserve interesting properties of the original action of $H$? We begin by formalizing this problem and present a construction of an induced action which behaves well when $H$ is hyperbolically embedded in $G$. Moreover, we show that induced actions can be used to characterize hyperbolically embedded subgroups. We also obtain some results for elementary amenable groups.